The canonical example of a polyadic algebra is an extension (generalization) of a functional monadic algebra, known as the functional polyadic algebra. Instead of looking at functions from XXX to AAA, we look at functions from XIsuperscriptXIX^{I} (where III is some set), the III-fold cartesian power of XXX, to AAA. In this entry, an element x∈XIxsuperscriptXIx\in X^{I} is written as a sequence of elements of AAA: (xi)i∈IsubscriptsubscriptxiiI(x_{i})_{{i\in I}} where xi∈AsubscriptxiAx_{i}\in A, or (xi)subscriptxi(x_{i}) for short.

Before constructing the functional polyadic algebra based on the sets X,IXIX,I and the Boolean algebra AAA, we first introduce the following notations:

for any J⊆IJIJ\subseteq I and x∈XIxsuperscriptXIx\in X^{I}, define the subset (of XIsuperscriptXIX^{I})

[x]J:={y∈XI∣xi=yi⁢ for every ⁢i∉J},assignsubscriptxJconditional-setysuperscriptXIsubscriptxisubscriptyi for every iJ[x]_{J}:=\{y\in X^{I}\mid x_{i}=y_{i}\mbox{ for every }i\notin J\},

for any function τ:I→Inormal-:τnormal-→II\tau:I\to I and any f:XI→Anormal-:fnormal-→superscriptXIAf:X^{I}\to A, define the function fτsubscriptfτf_{{\tau}} from XIsuperscriptXIX^{I} to AAA, given by

if f∈BfBf\in B, then fτ∈BsubscriptfτBf_{{\tau}}\in B for τ:I→Inormal-:τnormal-→II\tau:I\to I.

Note that if AAA were a complete Boolean algebra, we can take BBB to be AXIsuperscriptAsuperscriptXIA^{{X^{I}}}, the set of all functions from XIsuperscriptXIX^{I} to AAA.

Next, define ∃:P⁢(I)→BBnormal-:normal-→PIsuperscriptBB\exists:P(I)\to B^{B} by ∃(J)⁢(f)=f∃JJfsuperscriptfJ\exists(J)(f)=f^{{\exists J}}, and let SSS be the semigroup of functions on III (with functionalcompositions as multiplications), then we call the quadruple (B,I,∃,S)BIS(B,I,\exists,S) the functional polyadic algebra for the triple (A,X,I)AXI(A,X,I).

Remarks. Let (B,I,∃,S)BIS(B,I,\exists,S) be the functional polyadic algebra for (A,X,I)AXI(A,X,I).

(B,I,∃,S)BIS(B,I,\exists,S) is a polyadic algebra. The proof of this is not difficult, but involved, and can be found in the reference below.

If III is a singleton, then (B,I,∃,S)BIS(B,I,\exists,S) can be identified with the functional monadic algebra (B,∃)B(B,\exists) for (A,X)AX(A,X), for SSS is just III, and XIsuperscriptXIX^{I} is just XXX.

If III is ∅\varnothing, then (B,I,∃,S)BIS(B,I,\exists,S) can be identified with the Boolean algebra AAA, for S=∅SS=\varnothing and XIsuperscriptXIX^{I} is a singleton, and hence the set of functions from XIsuperscriptXIX^{I} to AAA is identified with AAA.